Bayes Theorem

Last updated Nov 1, 2022

Suppose $(\Omega, \mathcal{B}, \mathbb{P})$ is a Probability Space and $A, B \in \mathcal{B}$ so that $\mathbb{P}(B), \mathbb{P}(A) > 0$. Then

$$\mathbb{P}(B | A) = \frac{\mathbb{P}(A | B) \mathbb{P}(B)}{\mathbb{P}(A)}$$

Note that all values are Well-Defined by are assumption that $A,B$ are not Null Sets.

This follows from Definition 2 of Conditional Probability:

$$\mathbb{P}(B | A) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(A)} = \frac{\mathbb{P}(A | B) \mathbb{P}(B)}{\mathbb{P}(A)}$$

$\blacksquare$

• Conditional Probability